Estimating the Dynamics of Interest Rates in the Japanese Economy

Transcription

1 Estimating the Dynamics of Interest Rates in the Japanese Economy Professor K. Ben Nowman Westminster Business School University of Westminster 35 Marylebone Road London NW1 5LS, UK Dr. TrinoManuel Ñíguez Westminster Business School University of Westminster 35 Marylebone Road London NW1.5LS, UK August 8, 2006 Abstract This paper analyses the performance of the Yu and Phillips (2001) estimation method for the case of the Japanese economy. This new parametric technique is based on the Dambis, DurbinSchwarz theorem to extract an exact Gaussian discrete model of a continuoustime di usion process for the interest rate. We use the new approach together with Nowman s (1997) method to estimate di erent speci cations for a varied data set, including interbank and Tbill rates with di erent maturities and frequencies. Our results show that the Yu Phillips procedure provides estimates in line with their Monte Carlo results in most of the cases. Key words: Volatility; continuous time di usion; Monte Carlo JEL classi cation: C50, G10 We would like to thank Giorgio Di Pietro, Andreas Reschreiter, Antonio Rubia and participants at a Westminster Business School seminar and the Workshop in Memory of Rex Bergstrom, 2005 at WBS for valuable comments and suggestions. Responsibility for the contents remains solely with the authors.

2 1 Introduction Over the recent years there has been extensive econometric research on the estimation of continuous time di usion processes for the short term interest rate. The reason is that while those models are de ned in continuos time, the available data are always observed discretely in time, so ignoring such a di erence can result in an inconsistent estimator and, consequently, in wrong statistical inference (see, e.g. Melino (1994)). Thus, we nd in the literature di erent estimation methods de ned so that a continuous record of observations is not necessary, (see, e.g., Sundaresan (2000)). Empirical applications for Japan include, Tse (1995), Shoji and Ozaki (1996) and Hiraki and Takezawa (1997). In particular, Tse (1995) and Hiraki and Takezawa (1997) use the General Method of Moments (GMM) to estimate the parameters of the Chan, Karolyi, Longsta and Sanders (1992, CKLS hereafter) model and its special cases, 1 including, the OrnsteinUhlenbeck speci cation proposed by Vasicek (1977); the Cox, Ingersoll and Ross (1985,CIR hereafter) model and the Brennan and Schwartz (1980,BS hereafter). An alternative estimation approach based on the Gaussian estimation method of Bergstrom (1983, 1985, 1986, 1990) extended to nonlinear short rate di usions in Nowman (1997) has recently been used in empirical work. Speci cally, Nowman (1997) proposed a modi ed version of the exact discrete model suggested and explored for estimation in Phillips (1972) (see also Bergstrom (1984, Theorem 3, p.1167)) to estimate the CKLS model and its special cases. The Nowman (1997) estimator is a quasimaximum likelihood estimator (QMLE, hereafter), since the discrete process is not the true discrete model corresponding to the original di usion process. Recent applications and extensions to multifactors are considered in Nowman (1998, 2001, 2002, 2003, 2006) and Episcopos (2000). In a recent paper, Yu and Phillips (2001) (YP, hereafter) proposed a new parametric estimation technique obtained by the direct application of the Dambis, DurbinSchwarz theorem 1 See, e.g., Du e and Glynn (1997), and Kessler and Sorensen (1999) for further details on the use of the GMM estimation method for continuous time di usion models. 2

3 (DDB hereafter) (see Revuz and Yor, 1999). The DDB lemma allows one to extract an exact discrete model with Gaussian disturbances from the di usion process by selecting the data according to a time transformation. The model parameters can then be easily estimated by maximium likelihood. In YP they put forward an estimation algorithm which involves using the Nowman estimator and the approximate discrete model in an intermediate stage and then the YP transformed exact discrete model and exact Gaussian estimator. Monte Carlo (MC) evidence reported in YP indicated that the Nowman estimator provides very good estimates of the di usion term in nite samples but that the YP approach improves the estimation of the drift and mean reversion parameters in a range of frequencies. More recent MC evidence by Kawai and Maekawa (2003) on the YP method and the Nowman method indicated that the YP method is superior when the volatility is large (see also Lo (2005)). An illustrative practical application in YP for UK and US interest rates shows that both methods produce very similar estimates for the UK, but di erent ones for the US. Furthermore, for the US case the results are contrary to the ndings obtained from the MC experiment in YP. The aims of this paper are to provide further evidence on whether the YP MC results hold in practice, by applying the methodology to the known particular case of the Japanese economy. Our empirical study considers di erent interest rate speci cations, including the CKLS, CIR and BS, as well as a varied data set, including Interbank Rates and TBills with di erent maturity periods and frequencies. We nd that the YP procedure provides empirical results in line with their MC results in most of the cases. We also nd that the YP procedure produces a negative estimate of the long term mean in some cases compared to Nowman s approach. The paper is organized as follows. Section 2 deals with theoretical and practical considerations of the YP and Nowman (1997) approaches. Section 3 provides a descriptive analysis of the data. Section 4 presents and comments on the results, and nally the main conclusions are summarised in Section 5. 3

4 2 Estimation Methods: Theory and Implementation Consider the continuos time parametric di usion process for the short term interest rate, r (t), dr (t) = [ + r (t)] dt + r (t) db (t) (1) where B (t) is a standard Brownian motion, r (t) is recorded at time t = 0; ; 2; : : : ; T ; with measuring the time interval between the observations, 2 and are the parameters representing the drift and mean reversion, respectively; and are the volatility and elasticity parameters, respectively, which measure the relationship between the volatility and the level of rates. The Exact Gaussian Estimation technique of YP is obtained by the direct application of the DDB theorem. They show that the speci cation in equations (2), (3) and (4) is the exact discrete representation corresponding to the interest rate di usion process in equation (1) above, r(t j+1 ) = (eh j+1 1) + e h j+1 r(t j ) + M(h j+1 ); (2) t j+1 = t j + h j+1 ; (3) 8 9 < Z s = h j+1 = inf : s j 2 e 2(s ) r 2 (t j + )d a ; 8s = 1; 2; : : : : (4) ; 0 where M(h j+1 ) = B(a) N(0; a) for any xed constant a > 0, and ft j g j0 is an iterative sequence of time points starting with t 0 = 0, and moving acording to the step discrete variable fh j g j0. It is interesting to note that the above discrete time model is de ned to not use equally spaced observations, but a subsample of the points is calculated according to equations (3) and (4), so the data are selected more frequently when the level of interest rates is high, and hence the conditional volatility is higher. The practical implementation of the YP estimation procedure is outlined in 4 stages: 2 Note that the discrete time step is taken as 1, 1 and 1 for monthly, weekly and daily data, respectively

5 Stage 1) Estimate the Vasicek model (1977) (equation (5) below) by the ML method to obtain an estimate of the unconditional variance of the data, ba, 3 r(t + ) = (e 1) + e r(t) + ", " N(0; a) (5) Stage 2) Estimate the modi ed discrete model speci ed in Nowman (1997) in equations (6) and (7) below, corresponding to the Nowman (1997) approximation of the interest rate di usion process in equation (1), 4 r(t) = (e 1) + e r(t ) + (t), (6) (t)= t 1 N(0; 2 2 (e2 1)r 2 (t 1)): (7) A QMLE, ^ = (b; b ; b; b) can then be obtained by maximasing the Gaussian likelihood function. Stage 3) Set a,, as ba, b, b respectively, and condition on them in the next stage. The method is so de ned to be conditioned on Nowman s estimates, so it aims to improve the more critical estimates of,. Note that Nowman s estimates of, are shown to perform quite well in nite samples in YP MC study. Furthermore, given the nonstationarity presented by interest rate data, the ML procedure yields biased estimates of, (see Andrews, 1993). 5 Stage 4) Choose initial values of, to be the Nowman estimates, and the numerical maximisation 3 Note that the Vasicek model assumes that the variance of r(t) is contant over time, so the sample variance of b" is an estimate of the unconditional variance of r(t): 4 The Nowman continues time approximation to the stochastic equation (1) is given by dr (t) = [ + r (t)] dt + r (t) (s)db (t) ; s t (s + 1): So the conditional volatility remains unchanged over the unit intervals [s t (s + 1)): 5 It is worth pointing out that the procedure could also provide more accurate estimates of,, by using the information on the nonlinear dependence in the data. 5

6 of the log of the Gaussian likelihood function gives the YP estimates, 2 NX 6 max 4 1 r(t j+1 ) f;g 2 ln a (eh j+1 1) e h j+1 r(t j ) 2a j= (8) where N is the selected sample size, and h j+1, as given below, is a discretetime approximation to the process de ned in equation (4) ( ) sx h j+1 = min s j 2 e 2(s i) r 2 (t j + i) a. (9) i=1 The MC study performed by YP to analyse their estimator s nite properties provides the following results: Firstly, it is shown that the new technique compares quite well with the Nowman s (1997) method in estimating the di usion parameters of the CKLS model. Secondly, the YP estimator presents a signi cant smaller bias than Nowman s, and lastly, YP estimates are smaller for the drift and larger for the speed of reversion parameters. Furthermore, they test those results through an empirical application for UK and US interest rates, and they show that both methods produce very similar estimates for the UK but di erent for the US sample. They also nd for the US estimation results that they are contrary to their ndings obtained from the MC experiment. 3 The Data The interest rate data used in this study were obtained from Datastream. We use monthly and weekly interbank rates with maturities of one, two and six months over the period July 1986 to July 2005, for a total of 228 monthly, and 993 weekly observations. We also use monthly one and three month Tbills over the period July 1985 to July 2005, and weekly Tbills for the one month maturity over the same period, for a total of 240 monthly, and 1044 weekly observations. As can be seen in the data the rates have fallen substantially from the beginning of the data period. The e ect of the Bank of Japan s zero interest rate policy (ZIRP) in February

7 and the 2001 quantitative monetary easing policy give the series an interesting levelvolatility e ects. Tables 1 and 2 show the descriptive statistics of these series. We observe that the level data present very large autocorrelation coe cients, with a very low rate of decay, what indicates that the series present unit roots, irrespective of the frequency. The ADF test of the null hypothesis for the levels cannot be rejected at any standard signi cance level, con rming what we observe from the autocorrelations, but it is rejected for the rst di erences so the interest rate series are I(1). It is a well known fact that for series with units roots, the ML estimate in general, and the Nowman s estimate in particular, of the speed of reversion is signi cantly biased downwards, which results in an upward biased estimate of the unconditional mean. [Insert Tables 1 and 2 about here] 4 Estimation Results and Comparative Analysis In this section we now present the empirical results for the di erent short rate models given in Table 3 for Japan. In Tables 4 to 12 the ML estimates of the Vasiciek model, the Nowman estimates of the CKLS, CIR and BS models and the YP exact Gaussian estimates are presented. Consistent standard errors are presented in brackets below the parameter estimates. [Insert Tables 36 about there] Turning to the interbank monthly data results in Tables 46. For the one month rate for the CKLS model the Nowman estimate is = 0:4994 and signi cant, indicating some evidence of a relationship between the volatility of rates and level of rates close to the CIR process. This compares to the estimate in Episcopos (2000) using also monthly one month interbank rate over 1985 to 1998 of = 0:4143. There is weak evidence of mean reversion as was found in CKLS for the US and Nowman (1997) for the UK. In comparison to the YP estimates we nd 7

8 Nowman s estimates for the CKLS are not close for the drift or the mean reversion parameters. In particular, the YP estimate for the CKLS model for is 0:0254 which is smaller than Nowman s of 0:0397 and the speed of mean reversion from YP is 0:0835 which is larger than Nowman s estimate of 0:1194 in line with the YP MC results. The unconditional mean is estimated to be 0:332 percent for Nowman s method and 0:304 percent by YP. Turning to the CIR model the conclusions of the YP and Nowman s estimates are the same as the CKLS estimates. 6 Lastly, for the BS model the YP estimates are not the same as the Nowman estimates. The YP method gives a smaller and larger in line with their MC results. For the two month rate (Table 5), for the CKLS model the Nowman estimate is = 0:4621 and signi cant. There is weak evidence of mean reversion. In comparison to the YP estimates we nd Nowman s estimates are not close for the drift and the mean reversion parameters. In particular, the YP estimate for the CKLS model for is 0:1562 which is smaller than Nowman s of 0:0248 and the speed of mean reversion from YP is 0:0562 which is larger than Nowman s method of 0:1072 in line with the YP MC results. Turning to the CIR and BS models the results of the YP estimates and Nowman s estimates are di erent with the YP drift smaller than the Nowman s method, and the mean reversion coe cient of YP larger than the Nowman estimate in line with the MC results of YP. For the six month rate (see Table 6) for the CKLS model the Nowman estimate is = 0:6728 and signi cant. There is weak evidence of mean reversion. In comparison to the YP estimates we nd the same general conclusions as was found for the one and two month rates. The unconditional mean is estimated to be 0:078 percent for Nowman s method and 0:872 percent by YP. Turning to the CIR and BS models the results are the same as for the one and two month rates. [Insert Tables 79 about here] 6 Note that the CIR estimates converge to the CKLS ones. This is due to the Nowman estimate of (in the CKLS model) is not signi cantly di erent from, and YP estimation is conditioned on Nowman s. 8

9 Turning to the interbank weekly data results now. For the one month rate (Table 7) the CKLS model Nowman estimate is = 0:3970 and signi cant. There is weak evidence of mean reversion. In comparison to the YP estimates we nd Nowman s estimates are not close for the drift but are for the mean reversion parameter. In particular, the YP estimate for the CKLS model for is 0:0447 which is smaller than Nowman s of 0:0603. The speed of mean reversion from YP is 0:1214 which is marginally smaller than Nowman s method of 0:1199 and contrary to the MC results. Turning to the CIR model the results of the YP estimates and Nowman s estimates are not the same, with the YP drift smaller than Nowman s method and the mean reversion parameter of YP larger than the Nowman estimate in line with the MC results. Lastly for the BS model the YP estimates are not the same as the Nowman estimates with the YP giving a smaller and smaller which is contrary to the MC results. For the two month rate (Table 8) for the CKLS model the Nowman estimate is = 0:4964 and signi cant. In comparison to the YP estimates we nd Nowman s estimates are not close for the drift and mean reversion parameters. In particular the YP estimate for the CKLS model for is 0:0427 which is smaller than Nowman s of 0:0186. and the speed of mean reversion from YP is 0:0946 which is slightly larger than Nowman s method of 0:1108 in line with YP MC results. For the six month rate (Table 9) for the CKLS model the Nowman estimate is = 0:6654 and signi cant. In comparison to the YP estimates we nd Nowman s estimates are not close for the drift but are for the mean reversion parameter. In particular, the YP estimate for the CKLS model for is 0:0278 which is smaller than Nowman s of 0:0033 in line with the MC results. The speed of mean reversion from YP is 0:1287 which is slightly smaller than Nowman s method of 0:1256. Turning to the CIR and BS models the results of the YP estimates and Nowman s estimates are di erent. The YP gives a smaller and larger in line with the MC results. [Insert Tables 1012 about here] 9

10 Lastly turning to the Tbills rates. For the monthly one and three month rates (Tables 10 and 11, respectively) for the CKLS model the Nowman estimates are = 0:4322 and = 0:3076 and signi cant. There is evidence of weak mean reversion. For the weekly one month rate (Table 12) the CKLS model Nowman estimate is = 0:3506 and signi cant. Overall for the Tbills we nd that the YP estimates are generally di erent from Nowman (1997) and that the results are in line with the MC results. It is interesting to note that in all cases, the drift and reversion coe cients both are not statistically di erent from zero, so all models considered explain sucessfully the nonstationarity in mean feature of the data. This result is in line with the assumptions of very frequently used models for modelling the term structure of interest rates, including, the CIR variable rate process of Cox et al. (1980), and the Dothan s (1978) model. 7 [Insert Figures 16 about here] We also illustrate the nonequally spaced observations used in the exact estimation of YP. In the gures we plot the actual full time series and the vertical lines representing the sampling points ft j g actually used for a selection of rates. What we observe is that the higher the volatility the nner is the sampling speed (see YP). A general feature we observe with Japanese interest rates has been the major fall in the rates over the last ve or so years. This results in very low volatility and few points actually being sampled over those years as can be seen in the gures. In contrast, over the early part of the data we use the majority of the data points. This is to be expected from the results of YP. Finally we also illustrate some QuantileQuantile plots of the residuals of the CKLS model. As was found in YP we nd normality is reasonably induced. 7 See Nowman (1998) for an application of Dothan s (1978) and Cox et al. (1980) models to shortterm interest rates from di erent economies, including the Japanese economy. 10

11 5 Conclusions In this paper we have applied the exact Gaussian estimator of YP for continuous time short rate models using the YP exact discrete time Gaussian model to estimate the parameters. The YP method uses nonequispaced observations and a timechange transformation in the exact discrete model. We applied the YP estimation algorithm which involves using the Nowman estimator and approximate discrete model in an intermediate stage and then the YP transformed exact discrete time model and exact Gaussian estimator. Using weekly and monthly data on Japanese interest rates we nd in most cases that the YP procedure provides estimates which are di erent from the Nowman estimates and that the results are in line with their MC evidence for nite samples. We also nd that the YP procedure produces a negative estimate of the long term mean in some cases compared to Nowman s approach. 11

12 References [1] Andrews, D.W.K., 1993, Exactly medianunbiased estimation of rst order autoregressive/unit root models. Econometrica 61, [2] Bergstrom, A.R., 1983, Gaussian estimation of structural parameters in higherorder continuous time dynamic models. Econometrica 51, [3] Bergstrom, A.R., 1984, Continuous time stochastic models and issues of aggregation over time. In: Griliches, Z., Intriligator, M.D. (Eds.), Handbook of Econometrics, Vol. II (Elsevier Science, Amsterdam). [4] Bergstrom, A.R., 1985, The estimation of parameters in nonstationary higherorder continuous time dynamic models. Econometric Theory 1, [5] Bergstrom, A.R., 1986, The estimation of open higherorder continuous time dynamic models with Mixed Stock and Flow Data. Econometric Theory 2, [6] Bergstrom, A.R., 1990, Continuous Time Econometric Modelling (Oxford University Press, Oxford). [7] Brennan, M.J., and E.S. Schwartz, 1980, Analyzing convertible bonds. Journal of Financial and Quantitative Analysis 15, [8] Chan, K.C., Karolyi, G.A., Longsta, F.A. and A.B. Sanders, 1992, An empirical comparison of alternative models of the shortterm interest rate. Journal of Finance 47, [9] Cox, J.C., Ingersoll, J.E. and S.A. Ross, 1980, An analysis of variable rate loan contracts. Journal of Finance 35, [10] Cox, J.C., Ingersoll, J.E. and S.A. Ross, 1985, A theory of the term structure of interest rates., Econometrica 53, [11] Dothan, U.L., 1978, On the term structure of interest rates. Journal of Financial Economics 6,

13 [12] Du e, D. and P. Glynn, 1997, Estimation of continuoustime Markov processes sampled at random time intervals. Unpublished paper, Stanford University. [13] Episcopos, A., 2000, Further evidence on alternative continuous time models of the shortterm interest rate. Journal of International Financial Markets, Institutions and Money 10, [14] Hiraki, T. and N. Takezawa, 1997, How sensitive is shortterm Japanese interest rate volatility to the level of the interest rate? Economic Letters, 56, [15] Kawai, K. and K. Maekawa, 2003, A note on YuPhillips estimation for a continuous time model of a di usion process. Unpublished paper, Hiroshima University. [16] Kessler, M. and M. Sorensen, 1999, Estimating equations based on eigenfunctions for a discretely observed di usion. Bernoulli 5, [17] Lo, K.M., 2005 An evaluation of MLE in a model of the nonlinear continuoustime shortterm interest rate. Bank of Canada working paper, Canada. [18] Melino, A., 1994, Estimation of continuoustime models in nance. In: C.A. Sims (Ed.), Advances in Econometrics: Sixth World Congress, Vol. II (Cambridge University Press, Cambridge). [19] Nowman, K.B., 1997, Gaussian estimation of singlefactor continuous time models of the term structure of interest rates. Journal of Finance 52, [20] Nowman, K.B., 1998, Continuous time short rate interest rate models. Applied Financial Economics 8, [21] Nowman, K.B., 2001, Gaussian estimation and forecasting of multifactor term structure models with an application to Japan and the United Kingdom. Asia Paci c Financial Markets 8, [22] Nowman, K.B., 2002, The volatility of Japanese interest rates: evidence for certi cate of deposit and gensaki rates. International Review of Financial Analysis 11,

14 [23] Nowman, K.B., 2003, A note on gaussian estimation of the CKLS and CIR models with feedback e ects for Japan. Asia Paci c Financial Markets, 10, [24] Nowman, K.B., 2006, Continuous time interest rate models in Japanese xed income markets. In Batten, J.A., Fetherston, T.A., Szilagyi P.G. (Eds.), Japanese Fixed Income Markets: Money, Bond and Interest Rate Derivatives (Elsevier Science, Amsterdam). [25] Revuz, D. and M. Yor, 1999, Continuous martingales and brownian motion (Springer, New York). [26] Shoji, I. and T. Ozaki, 1996, A statistical comparison of the shortterm interest rate models for Japan, U.S. and Germany. Asia Paci c Financial Markets 3, [27] Sundaresan, S.M., 2000, Continuoustime methods in nance: a review and an assessment. Journal of Finance 55, [28] Phillips, P.C.B., 1972, The structural estimation of a stochastic di erential equation system. Econometrica, 40, [29] Tse, Y.K., 1995, Some international evidence on the stochastic behavior of interest rates. Journal of International Money and Finance, 14, [30] Yu, J. and P.C.B. Phillips, 2001, A gaussian approach for continuous time models of the shortterm interest rate. Econometrics Journal, 4, [31] Vasicek, O., 1977, An equilibrium characterization of the term structure. Journal of Financial Economics 5,

15 Table 1: Deescriptive Statistics: Interbank Rates Variable Mean S.D ADF Interbank Monthly 1 month r (t) 2:3584 2:5777 0:99 0:98 0:97 0:95 0:94 0:92 1:0680 r (t) 0:0207 0:2298 0:14 0:03 0:19 0:09 0:04 0:10 3: month r (t) 2:3696 2:5791 0:99 0:97 0:96 0:95 0:94 0:92 1:1111 r (t) 0:0201 0:1995 0:11 0:10 0:06 0:06 0:05 0:14 4: month r (t) 2:3777 2:5527 0:99 0:98 0:97 0:95 0:94 0:92 0:9875 r (t) 0:0185 0:1381 0:28 0:16 0:11 0:08 0:12 0:12 7:4266 Interbank Weekly 1 month r (t) 2:3720 2:5777 0:99 0:99 0:99 0:98 0:98 0:98 1:0177 r (t) 0:0048 0:1271 0:08 0:03 0:11 0:21 0:03 0:04 5: month r (t) 2:3803 2:5782 0:99 0:99 0:99 0:99 0:98 0:98 0:5552 r (t) 0:0046 0:0885 0:08 0:11 0:01 0:04 0:02 0:00 30: month r (t) 2:3862 2:5509 0:99 0:99 0:99 0:99 0:98 0:98 0:5299 r (t) 0:0042 0:057 0:16 0:14 0:08 0:04 0:05 0:07 27:

16 Table 2: Descriptive Statistics: Tbill Rates Variable Mean S.D ADF Tbill Monthly 1 month r (t) 2:4917 2:6382 0:98 0:97 0:95 0:94 0:92 0:89 1:6741 r (t) 0:0192 0:2964 0:14 0:09 0:02 0:05 0:14 0:42 7: month r (t) 2:4956 2:6203 0:98 0:97 0:95 0:94 0:91 0:89 2:2115 r (t) 0:0189 0:3034 0:16 0:21 0:20 0:02 0:11 0:08 6:9405 Tbill Weekly 1 month r (t) 2:5002 2:6347 0:99 0:99 0:99 0:98 0:98 0:97 1:1407 r (t) 0:0044 0:2059 0:33 0:02 0:06 0:11 0:01 0:02 36:

17 Table 3: Short Rate Interest Rate Models Name CKLS (1992) Vasicek (1977) CIR (1985) BS (1980) Model dr (t) = f + r (t)g dt + r db dr (t) = f + r (t)g dt + db dr (t) = f + r (t)g dt + r 1 2 db dr (t) = f + r (t)g dt + rdb 17

18 Table 4: Gaussian Estimates Interbank Rate: Monthly One Month Model Estimation Method (a) Log LF Vasicek ML (0.1405) (0.0960) CKLS Nowman (0.0412) (0.0836) (0.0554) (0.0811) CKLS YuPhillips (0.1009) (0.0924) CIR Nowman (0.0466) (0.0841) (0.0670) CIR YuPhillips (0.1009) (0.0924) BS Nowman (0.0251) (0.1917) (0.3628) BS YuPhillips (0.0594) (0.0919)

19 Table 5: Gaussian Estimates Interbank Rate: Monthly Two Month Model Estimation Method (a) Log LF Vasicek ML (0.1239) (0.0800) CKLS Nowman (0.0389) (0.0668) (0.0426) (0.0794) CKLS YuPhillips (0.1478) (0.0782) CIR Nowman (0.0449) (0.0663) (0.0576) CIR YuPhillips (0.1718) (0.0823) BS Nowman (0.0273) (0.1545) (0.3106) BS YuPhillips (0.0829) (0.0764)

20 Table 6: Gaussian Estimates Interbank Rate: Monthly Six Month Model Estimation Method (a) Log LF Vasicek ML (0.1193) (0.0826) CKLS Nowman (0.0183) (0.0703) (0.0285) (0.0615) CKLS YuPhillips (0.0916) (0.0788) 711 CIR Nowman (0.0262) (0.0671) (0.0259) CIR YuPhillips (0.1500) (0.0800) BS Nowman (0.0123) (0.1168) (0.0583) BS YuPhillips (0.0635) (0.0794)

21 Table 7: Gaussian Estimates Interbank Rate: Weekly One Month Model Estimation Method (a) Log LF Vasicek ML (0.1645) (0.0911) CKLS Nowman (0.0624) (0.0802) (0.0351) (0.1013) CKLS YuPhillips (0.0620) (0.0839) CIR Nowman (0.0653) (0.0824) (0.1350) CIR YuPhillips (0.0570) (0.0816) BS Nowman (0.0408) (0.0388) (0.6632) BS YuPhillips (0.0851) (0.0890)

22 Table 8: Gaussian Estimates Interbank Rate: Weekly Two Month Model Estimation Method (a) Log LF Vasicek ML (0.1162) (0.0726) CKLS Nowman (0.0313) (0.0632) (0.0171) (0.0808) CKLS YuPhillips (0.0979) (0.0748) CIR Nowman (0.0384) (0.0631) (0.0437) CIR YuPhillips (0.0607) (0.0662) BS Nowman (0.0231) (0.1444) (0.2618) BS YuPhillips (0.1387) (0.0735)

23 Table 9: Gaussian Estimates Interbank Rate: Weekly Six Month Model Estimation Method (a) Log LF Vasicek ML (0.1046) (0.0671) CKLS Nowman (0.0168) (0.0593) (0.0078) (0.0393) CKLS YuPhillips (0.0882) (0.0669) CIR Nowman (0.0234) (0.0555) (0.0152) CIR YuPhillips (0.0423) (0.0579) BS Nowman (0.0122) (0.1038) (0.0322) BS YuPhillips (0.1475) (0.0662)

24 Table 10: Gaussian Estimates Tbill Rate: Monthly One Month Model Estimation Method (a) Log LF Vasicek ML (0.1651) (0.1059) CKLS Nowman (0.0615) (0.0821) (0.0567) (0.0661) CKLS YuPhillips (0.0500) (0.0907) CIR Nowman (0.0696) (0.0821) (0.0764) CIR YuPhillips (0.0345) (0.0940) BS Nowman (0.1076) (0.3928) (0.3784) BS YuPhillips (0.1005) (0.1023)

25 Table 11: Gaussian Estimates Tbill Rate: Monthly Three Month Model Estimation Method (a) Log LF Vasicek ML (0.1705) (0.0885) CKLS Nowman (0.0872) (0.0726) (0.0780) (0.0945) CKLS YuPhillips (0.0768) (0.0779) CIR Nowman (0.0978) (0.0742) (0.1654) CIR YuPhillips (0.0931) (0.0764) BS Nowman (0.0691) (0.2180) (0.7027) BS YuPhillips (0.2348) (0.0930)

26 Table 12: Gaussian Estimates Tbill Rate: Weekly One Month Model Estimation Method (a) Log LF Vasicek ML (0.2173) (0.0974) CKLS Nowman (0.0946) (0.0811) (0.0292) (0.0443) CKLS YuPhillips (0.1092) (0.0841) CIR Nowman (0.0820) (0.0841) (0.0710) CIR YuPhillips (0.1094) (0.0914) BS Nowman (0.0940) (0.4419) (0.1945) BS YuPhillips (388) (0.1162)

27 /90 1/95 1/00 1/05 Figure 1: Time Transformations for Interbank : Monthly 1 Month /90 1/95 1/00 1/05 Figure 2: Time Transformations for Interbank : Monthly 2 Month 27

28 /90 1/95 1/00 1/05 Figure 3: Time Transformations for Interbank : Monthly 6 Month /90 1/95 1/00 1/05 Figure 4: Time Transformations for Tbill : Monthly 3 Month 28

29 3 Theoretical QuantileQuantile 2 Normal Quantile Figure 5: QQ Plot of Residuals : Interbank Monthly 1 Month 3 Theoretical QuantileQuantile 2 Normal Quantile Figure 6: QQ Plot of Residuals : Interbank Weekly 1 Month 29